Constructing Permutation Arrays from Semilinear Groups Avi Levy 2/17/2013 Contents 1 Introduction 2 2 Overview and Notation 2 2.1 Finite Fields . 2 2.2 Permutation Polynomials . 2 2.3 Classical Groups . 2 2.4 Hamming distance and Permutation Arrays . 3 3 Constructions 3 3.1 AGL(1; Fq) ............................ 3 3.2 AΓL(1; Fq)............................. 3 3.3 P GL(2; Fq) ............................ 4 3.4 P ΓL(1; Fq)............................. 4 4 Hamming Distance Computations 4 4.1 Preliminaries . 4 4.2 AΓL ................................ 5 4.3 P ΓL ................................ 6 5 Root-counting Results 7 6 Conclusions 10 7 Bibliography 10 1 1 Introduction This is a preliminary writeup of my work with the semilinear groups. These groups yield new permutation arrays with large pairwise Hamming distances. Their properties are well-understood, which allows the minimal pairwise Hamming distance to be obtained without resorting to direct computation. 2 Overview and Notation 2.1 Finite Fields n Given q = p , a power of a prime, call the unique field with q elements Fq. 2 q−2 Fq = f0; 1; g; g ; ··· ; g g where g is any generator of Fq. 2.2 Permutation Polynomials Let f : Fq ! Fq be a polynomial function with coefficients in Fq. If f is one-to-one, then f permutes the elements of Fq. In this case, f is called a permutation polynomial, and the permutation corresponding to f is: σf : x ! f(x) 2.3 Classical Groups There are four families of groups that we are concerned with in this paper. • AGL(n; F): affine general linear group of dimension n over the field F. • AΓL(n; F): affine general semilinear group of dimension n over F. • P GL(n; F): projective general linear group of dim n over F. • P ΓL(n; F): projective general semilinear group of dim n over F. Note 2.1. We will use shorthand like Group(n; q) as a synonym for Group(n; Fq). 2 2.4 Hamming distance and Permutation Arrays In this section, all permutations act on n elements. The Hamming distance between two permutations is the number of places in which they differ. Let S and T be sets of permutations. Then hd(S; T ) denotes the minimal Hamming distance between distinct elements of S and T . If hd(S; S) = d, then the set S is called a permutation array of Hamming distance d. We say that S is an M(n; d) in this case. 3 Constructions 3.1 AGL(1; Fq) AGL(1; Fq) = AGL(1; q) is the group of linear polynomials: AGL(1; q) = fax + bja; b 2 Fq; a 6= 0g The group operation is function composition: (ax + b) ◦ (cx + d) = a(cx + d) + b = acx + (b + d) In fact, all the groups in this paper are presented as sets of polynomials, and the group operation will always be function composition. This group is sharply 2−transitive and yields an optimal M(q; q−1) with q(q−1) elements. 3.2 AΓL(1; Fq) Recall that q = pn. Consider the permutation polynomial frob(x) = xp called the Frobenius automorphism, which is semilinear in the following sense: frob(x + y) = (x + y)p ≡ xp + yp = frob(x) + frob(y) Starting from AGL(1; q), append frob(x) and take the group closure, yielding: pi AΓL(1; q) = fax + bja; b 2 Fq; a 6= 0; 0 ≤ i < ng This group has nq(q − 1) elements. 3 3.3 P GL(2; Fq) Projective groups act on a \point at infinity". To accommodate this case, 1 form the set P (Fq) = f1g [ Fq. Then P GL(2; q) is constructed as: ax + b P GL(2; q) = a; b; c; d 2 Fq; ad 6= bc cx + d These are called fractional linear functions. Note that cancelling common factors from the numerator and denominator leaves the function unchanged, so there are only 3 degrees of freedom among (a; b; c; d). ax+b Suppose h(x) 2 P GL(2; q). If h(x) is written as h(x) = cx+d , then h acts 1 on P (Fq) as follows: 8 a c if x = 1 < d h(x) = 1 if x = − c : ax+b cx+d otherwise The group P GL(2; q) is 3−transitive and yields an optimal M(q + 1; q − 1) with (q + 1)q(q − 1) elements. 3.4 P ΓL(1; Fq) In analogy with AΓL, start from P GL(2; q) then append frob(x) and take the group closure. ( i ) axp + b P ΓL(2; q) = a; b; c; d 2 ; ad 6= bc; 0 ≤ i < n pi Fq cx + d This group has n(q + 1)q(q − 1) elements. 4 Hamming Distance Computations 4.1 Preliminaries In this section, all permutations act on n elements. Let G be a set of permutations that is also a permutation group. The goal in this section is to compute hd(G; G), and thereby interpret G as a permutation array M(n; d). 4 Lemma 4.1. hd(fστg; fσρg) = hd(fτg; fρg) Proof. στ(x) = σρ(x) () τ(x) = ρ(x) Lemma 4.2. hd(G; G) = hd(feg;G), where e is the identity permutation Proof. Pick a; b 2 G. Then hd(a; b) = hd(e; a−1b) so the result follows Lemma 4.3. Suppose N is a normal subgroup of a finite group, G. If faigi2I is a set of coset representatives of N, then hd(G; G) = min hd(faig;N) i2I −1 Proof. Since N is normal, the set fai g is also a set of coset representatives. S −1 Therefore, G = i2I ai N, so the Hamming distance is computed as follows: −1 hd(G; G) = hd(feg;G) = min hd(feg; ai N) = min hd(faig;N) i2I i2I where Lemma 4.1 and Lemma 4.2 have been applied. Using the following lemma, one can compute the Hamming distance sim- ply by counting roots of polynomials. Lemma 4.4. For any polynomial f 2 Fq[x], let r(f) denote the number of distinct roots of f(x) in Fq. Then for any distinct pair of polynomials g; h 2 Fq[x] we have hd(fgg; fhg) = n − r(g − h) Proof. g(x) 6= h(x) is equivalent to saying x is not a root of g − h 4.2 AΓL Theorem 4.5. AΓL(1; q) is an M q; q − pn∗ , where n∗ denotes the largest proper factor of n. Proof. Let G = AΓL(1; q) and let N = AGL(1; q). Note that N is normal in G by construction. To proceed, apply Lemma 4.3 with representatives pi n−1 fx gi=0 to G, then use Lemma 4.4: hd(G; G) = min hd(fxpi g;N) 0≤i<n i = q − max r xp − g(x) i<n g2N 5 By Theorem 5.1, i i max r xp − g(x) = r xp − x g2N Now by Theorem 5.2, i r xp − x = pgcd(i;n) Putting all of our results together, i hd(G; G) = q − max r xp − g(x) i<n g2N = q − max pgcd(i;n) i<n = q − pn∗ Thus AΓL(1; q) is an M q; q − pn∗ . Corollary 4.6. Suppose q = 2p where p is prime. Then there exists an M(q; q − 2) of size pq(q − 1). 4.3 P ΓL Theorem 4.7. P ΓL(2; q) is an M q + 1; q − pn∗ , where n∗ denotes the largest proper factor of n. Proof. Let G = P ΓL(2; q) and H = fg 2 Gjg(1) = 1g. H can be identified with AΓL(1; q) by means of the isomorphism i : AGL(1; q) ! H g(x) if x 2 i(g)(x) = Fq 1 if x = 1 Suppose g(x); h(x) are distinct elements of G. If g(x) 6= h(x) for all x, then hd(fgg; fhg) = q + 1. Otherwise, pick a point y such that g(y) = h(y) = w Choose the following elements of G: ( 1 if w 2 Fq τ1(x) = x − w x if w = 1 6 ( yx + 1 if y 2 Fq τ2(x) = x x if y = 1 0 0 Let g = τ1 ◦ g ◦ τ2; h = τ1 ◦ h ◦ τ2. Then τ1(g(τ2(1))) = 1 and likewise for h, so g0; h0 2 H. By Lemma 4.1, hd(g0; h0) = hd(g; h). Moreover, the isomorphism i pre- serves Hamming distance so hd(g0; h0) = hd i−1(g0); i−1(h0) But i−1(g0); i−1(h0) 2 AΓL(1; q), so by Theorem 4.7 hd i−1(g0); i−1(h0) ≥ q − pn∗ Therefore hd(g; h) ≥ q − pn∗ , so P ΓL(2; q) is an M q + 1; q − pn∗ . Corollary 4.8. Let q = 2p where p is prime. Then there exists an M(q + 1; q − 2) of size p(q + 1)q(q − 1) = O (q3 log q). This improves upon the best current computational results. 5 Root-counting Results Theorem 5.1. r xpi + ax + b ≤ r xpi − x pi pi pi Proof. Let p1(x) = x + ax + b; p2(x) = x + ax and p3(x) = x − x. We will show that r(p1) ≤ r(p2) ≤ r(p3). First, suppose p1 has at least one root (if not, the result holds trivially). Then pick a root of p1 and call it y. Observe that for any root yi of p1, we have pi p2(yi − y) = (yi − y) + a(yi − y) pi pi = yi − y + a(yi − y) = p1(yi) − p1(y) = 0 Thus y1 −y is a root of p2.